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8.7 Volumes with Cross Sections: Squares and Rectangles

Part of Applications of Integration in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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If a solid has a known base region and known cross sections perpendicular to an axis, then its volume can be found by integrating the area of each cross section. For squares and rectangles, you first express the side length or dimensions of a typical cross section in terms of x or y, then use the appropriate area formula, and finally integrate across the interval. In general, the volume formula is
V=∫abA(x) dxV=\int_a^b A(x)\,dx
or
V=∫cdA(y) dyV=\int_c^d A(y)\,dy
where A is the cross-sectional area.

The main challenge is identifying the dimension of the cross section from the base region. If the cross section is a square, then its area is side squared. If it is a rectangle, you may need a fixed ratio or given height relationship. You should think of this as a three-dimensional version of slicing: each slice has a little area, and integrating adds those cross-sectional areas into a total volume. This method is different from disc and washer methods because the solid is not formed by rotation; instead, the shape of the cross section is given directly.

In this unit

13 lessons

  1. 8.1Finding the Average Value of a Function on an IntervalPreview lesson
  2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using IntegralsPreview lesson
  3. 8.3Using Accumulation Functions and Definite Integrals in Applied ContextsPreview lesson
  4. 8.4Finding the Area Between Curves Expressed as Functions of xPreview lesson
  5. 8.5Finding the Area Between Curves Expressed as Functions of yPreview lesson
  6. 8.6Finding the Area Between Curves That Intersect at More Than Two PointsPreview lesson
  7. 8.7Volumes with Cross Sections: Squares and RectanglesThis lesson
  8. 8.8Volumes with Cross Sections: Triangles and SemicirclesPreview lesson
  9. 8.9Volume with Disc Method: Revolving Around the x- or y-AxisPreview lesson
  10. 8.10Volume with Disc Method: Revolving Around Other AxesPreview lesson
  11. 8.11Volume with Washer Method: Revolving Around the x- or y-AxisPreview lesson
  12. 8.12Volume with Washer Method: Revolving Around Other AxesPreview lesson
  13. 8.13The Arc Length of a Smooth, Planar Curve and Distance TraveledPreview lesson

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